200 15 Viscoelastic Composites

15.1 Composite Analysis

15.1.1 Accurate Analysis

The theoretical basis of the analysis presented in this chapter is the findingof Hashin’s [165,166] that the complex stiffness (EC) of a composite made ofviscoelastic phases (P, S) can be determined by the elastic counterpart com-posite stiffness (E), replacing phase stiffness (EP , ES) with their respectivecomplex counterparts (EPC , ESC).

Keeping in mind the significance of the analogy Young’s modulus previ-ously introduced in Sect. 14.1.1, and re-calling the vibration analogy fromSect. 14.1.2, Hashin’s observation means that the analogy Young’s modulusof a composite material (EA) can be obtained from the elastic counterpartstiffness solution (E), replacing phase stiffness (EP , ES) with their respectiveanalogy Young’s moduli (EA

P, EA

S).

Having established the basic rheological property (EA) of the compositematerial we may proceed just as explained in Chap. 14 with the compositematerial considered as a homogeneous viscoelastic material. A further conclu-sion which can be made from [165,166] and the concept presented in this bookof an analogy Young’s modulus, is that composite stress/strain solutions canbe established from Laplace transforming the elastic counterpart solutions(Table 10.3) and replacing (EP , ES) with (EA

P, EA

S). A summary of results

such obtained from an analysis of composite materials is presented in Tables15.1 and 15.2. Analogy Young’s moduli (EA

P, EA

S) for various homogeneous

materials are presented in Sect. 14.2.

Remarks: The procedure presented in Table 15.1 of determining the creepand relaxation functions for a composite material from the complex stiff-ness is very efficient using computers with the capability of handling complexnumbers. It is obvious how the experimental vibration analysis explained inSect. 14.1.2 can be used also in the research on composite geometry versusviscoelasticity of composite materials.

With respect to the determination of creep- and relaxation functions were-call that only one of these functions are needed to predict the other one bythe basic (14.2).

In general the determination of internal stress and eigenstrain-stress asexpressed in Table 15.2 calls for numerical Laplace-inversions. Alternatively,approximate solutions can be found as explained in the following section.

15.1.2 Approximate Analysis

A very simple approximate quasi-static analysis of viscoelastic composites canbe obtained from the elastic counterpart analysis explained in (15.1),

F ≈ FELAST

(

P,EEFF

P , EEFF

S

)

where FELAST = FELAST(P, EP , ES) (15.1)

15.1 Composite Analysis 201

Table 15.1. Complex Young’s modulus, creep, and relaxation of composite materialdetermined from Young’s modulus of such material. Examples are composites madeof components exhibiting Power law viscoelasticity. [ ]P and [ ]S mean that E, b, andτ in [ ] are subscripted as indicated

Young’s Modulus ⇒ Analogy Young’s Modulus

E = E(EP , ES) ⇒ EA = EA(s) = E(EA

P , EA

S )

Example EA

P =

[

E(τs)b

Γ(1 + b) + (τs)b

]

P

; EA

S =

[

E(τs)b

Γ(1 + b) + (τs)b

]

S

Analogy Young’s Modulus ⇒ Complex Young’s Modulus

EA = EA(s) = E(EA

P , EA

S ) ⇒ EC = EA(iω) = E(EPC , ESC) = ER(ω) + iEI(ω)

Example EPC =

[

E(iτω)b

Γ(1 + b) + (iτω)b

]

P

; ESC =

[

E(iτω)b

Γ(1 + b) + (iτω)b

]

S

Complex Young’s Modulus ⇒ Relaxation and Crrep

R(t) = E − 2π

∞∫

0

EI(ω)1 − cos(ωt)

ω dω

C(t) = 1E + 2

π

∞∫

0

JI(ω)1 − cos(ωt)

ω dω;

(

JI = EI

|EC |2

)

Or Analogy Young’s Modulus ⇒ Relaxation and Creep

C(t) = L−1

[

1sEA

]

; R(t) = L−1

[

EA

s

]

Table 15.2. Internal stress from external load on, and eigenstrain/stress in com-posite material determined from analogy Young’s modulus of such material

Internal Stress from Ext-Load (eA = EA/EA

S; nA = EA

P /EA

S )

σP (t) = 1c L−1

[

σ1/eA − 1

1/nA − 1

]

; σS(t) =σ(t) − cσP (t)

1 − c

Eigenstrain/Stress (KA

S ≈ EA

S /1.8)

λ(t) = L−1

(

λS + ∆λ1/eA − 1

1/nA − 1

)

with ∆λ = λP − λS

ρP (t) = −L−1

(

3∆λKA

S

c(1/nA − 1) − (1/eA − 1)

c(1/nA − 1)2

)

; ρS(t) = − c1 − cρP (t)

202 15 Viscoelastic Composites

where FELAST and F are the elastic and viscoelastic solutions respectivelyto the problem considered. EEFF

Pand EEFF

Sare the effective Young’s moduli

explained in Sect. 14.3.1. Load (stress or strain) is denoted by P . In anotherformulation the method was first suggested by Ross [158] as an easy way ofestimating the stress distribution in a composite structure made of concreteand steel. An example of applying (15.1) in a composite analysis is explainedin (15.2). The elastic composite stiffness, E, is converted to the creep functionof the counterpart viscoelastic composite.

C(t) ≈1

EEFFwith EEFF = E

(

EEFF

P , EEFF

S

)

where E = E(EP , ES) (15.2)

Remark: The quality of composite solutions obtained by the “compositeEEFF-method” depends on the quality of EEFF

Pand EEFF

Sconsidered in Sect.

14.3.1. The load restrictions explained in this section must hold for each phasealso on a “composite level”. The author’s (tentative) experience with respectto the quality of EEFF-estimates is the following:

Estimates of “reasonable accuracy” can be obtained for the material prop-erties, creep functions, creep stresses, eigenstress/strain properties. To geta similar level of accuracy for “estimated” relaxation functions, it might benecessary to determine this function numerically from the creep functionusing the basic (14.2).In general, estimates of reasonably high accuracy can be expected in anyanalysis when composites are considered where both components have PowerLaw viscoelasticity with b < 1/3.

These statements are based on testing the composite EEFF-method on com-posites such as Maxwell materials mixed with elastic spheres, mixtures of twomaterials exhibiting Power Law creep, and layered composites made of twoMaxwell materials.

Approximate Inversion Method

It is tempting (see Sect. 14.3.1) to use this method when problem solutions areformulated by their Laplace transformed as they are in Tables 15.1 and 15.2:Multiply the Laplace transformed solution with s and then replace s with γ/t.The inversion parameter γ, however, has to be estimated as some compositeaverage of inversion parameters applying to phases P and S. In the author’sopinion this feature disqualifies, in practice, the approximate inversion methodto be better than the plain EEFF-method. In any case, more research has tobe made on this matter.

15.2 Applications

Some examples are presented in this section, which illustrate how the analy-sis of viscoelastic composites just explained in Sect. 15.1 works on various